M1 June 2017 Q3
3. A plank \(AB\) has length 6 m and mass 30 kg. The point \(C\) is on the plank with \(CB = 2\) m. The plank rests in equilibrium in a horizontal position on supports at \(A\) and \(C\). Two people, each of mass 75 kg, stand on the plank. One person stands at the point \(P\) of the plank, where \(AP = x\) metres, and the other person stands at the point \(Q\) of the plank, where \(AQ = 2x\) metres. The plank remains horizontal and in equilibrium with the magnitude of the reaction at \(C\) five times the magnitude of the reaction at \(A\). The plank is modelled as a uniform rod and each person is modelled as a particle.
| Scheme | Marks |
|---|---|
| \((\uparrow)\ \ R + 5R = 75g + 30g + 75g\) | M1 A2 |
| \(M(A)\ \ \ 75gx + 75g2x + 30g \times 3 = 5R \times 4\) | M1 A2 |
| \(x = \tfrac{34}{15} = 2.3\) or better | A1 |
| ( N.B. Or another Moments Equation) | (M1 A2) |
| (7) |
Notes
First M1 for either a vertical resolution (with correct of terms) or a moments equation (all terms dim correct and correct no. of terms)
First A1 and Second A1 for a correct equation in \(R\) (or \(S\) where \(S = 5R\)) only or \(R\) and \(x\) only or \(S\) and \(x\) only. (−1 each error, A1A0 or A0A0)
Second M1 for a moments equation (all terms dim correct and correct no. of terms)
Third A1 and Fourth A1 for a correct equation in \(R\) (or \(S\) where \(S = 5R\)) only or \(R\) and \(x\) only or \(S\) and \(x\) only. (−1 each error, A1A0 or A0A0)
Fifth A1 for \(x = 34/15\) oe or 2.3 (or better)
(i) In a moments equation, if \(R\) and \(5R\) (or \(S\) and \(0.2S\)) are interchanged, treat as 1 error.
(ii) Ignore diagram if it helps the candidate.
(iii) If an equation is correct but contains both \(R\) and \(S\), or \(S = 5R\) is never used, treat as 1 error.
(iv) Full marks possible if all \(g\)’s omitted.
(v) For inconsistent omission of \(g\), penalise each omission.
\(M(B),\ R \times 6 + 5R \times 2 = 75g(6 - x) + 75g(6 - 2x) + 30g \times 3\)
\(M(C),\ 75g(4 - x) + 75g(4 - 2x) + 30g \times 1 = R \times 4\)
\(M(G),\ 75g(3 - x) + 5R \times 1 = R \times 3 + 75g(2x - 3)\)
\(M(P),\ Rx + 30g(3 - x) + 75gx = 5R(4 - x)\)
\(M(Q),\ 75gx + 30g(2x - 3) + 5R(4 - 2x) = R \times 2x\)
| Scheme | Marks |
|---|---|
| uniform – mass is or acts at midpoint of plank; centre of mass is at middle of plank; weight acts at the middle of the plank, centre of gravity is at midpoint | |
| rod - plank does not bend, remains straight, is inflexible, is rigid | B1 B1 |
| (2) | |
| (9 marks) |
Notes
First B1 for first correct answer seen.
Second B1 for the other answer, but only award this second mark if no extras given.